For a scene in a movie, a stunt driver drives a 1.90 103 kg pickup truck with a length of 4.09 m around a circular curve with a radius of curvature of 0.333 km. The truck is to curve off the road, jump across a gully 10.0 m wide, and land on the other side 2.96 m below the initial side. What is the minimum centripetal acceleration the truck must have in going around the circular curve to clear the gully and land on the other side?
well, you need the initial vertical velocity approaching the ditch.
over the ditch, one drops 2.96m. How long does it take to fall that distance?
2.96=1/2 g t^2 or t= sqrt(2*2.96/9.8)=.9455 sec check that.
so, to get across the ditch in this time, velocity= 10m/.9455=10.575m/s
centripetal acceleration= v^2/r
To find the minimum centripetal acceleration the truck must have in going around the circular curve, we can use the concept of conservation of energy.
Step 1: Calculate the initial potential energy of the truck.
The truck starts at a height of 2.96 m above the final landing point. We can calculate the initial potential energy using the mass of the truck (m = 1.90 * 10^3 kg) and the acceleration due to gravity (g = 9.8 m/s^2):
Initial Potential Energy = mass * height * gravity
= 1.90 * 10^3 kg * 2.96 m * 9.8 m/s^2
Step 2: Calculate the final potential energy of the truck.
The truck lands 2.96 m below the initial height. Therefore, the final potential energy is zero.
Step 3: Calculate the change in potential energy.
Change in Potential Energy = Final Potential Energy - Initial Potential Energy
= 0 - (1.90 * 10^3 kg * 2.96 m * 9.8 m/s^2)
Step 4: Calculate the kinetic energy required to clear the gully.
The vertical velocity at the start and end of the jump is zero. Therefore, the change in kinetic energy is equal to the change in potential energy.
Change in Kinetic Energy = Change in Potential Energy
Step 5: Calculate the centripetal acceleration using kinetic energy.
The kinetic energy is given by the formula:
Kinetic Energy = (1/2) * mass * velocity^2
But we know that the transition from potential energy to kinetic energy is equal to the change in potential energy calculated above. Therefore:
Change in Kinetic Energy = (1/2) * mass * velocity^2
= Change in Potential Energy
Given that the radius of curvature of the circular curve is 0.333 km, we can convert it to meters (since all other measurements are in meters):
Radius = 0.333 km = 0.333 * 10^3 m
Step 6: Calculate the minimum centripetal acceleration.
The minimum centripetal acceleration required to clear the gully can be calculated using the formula:
Centripetal Acceleration = (velocity^2) / radius
Setting the change in kinetic energy equal to the change in potential energy, we can solve for the velocity:
(1/2) * mass * velocity^2 = Change in Potential Energy
Once we have the velocity, we can substitute it into the formula for centripetal acceleration and solve for the minimum centripetal acceleration.
Note: This calculation assumes no air resistance or friction during the jump. In reality, additional factors would need to be considered for a more accurate calculation.